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Chromonic liquid crystals are lyotropic nematic phases whose applications span from food to drug industries. It has recently been suggested that the elastic energy density governing the equilibrium distortions of these materials may be quartic in the measure of twist. Here we show that the non-linear twist-wave equation associated with such an energy has smooth solutions that break down in a finite time, giving rise to the formation of a shock wave, under rather generic assumptions on the initial profile. The critical time at which smooth solutions become singular is estimated analytically with an accuracy that numerical calculations for a number of exemplary cases prove to be satisfactory.

Minimal surfaces are ubiquitous in nature. Here they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that (1) every minimal surface is transformed into a minimal surface by a bending-neutral deformation, (2) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other, and (3) all minimal surfaces have indeed a universal bending content.

Twisted hedgehogs are defects in spherical cavities with homeotropic anchoring for the nematic director that arise when twist distortions are sufficiently less energetic than splay (and bend) distortions. They bear a characteristic inversion ring, where the director texture changes the sense it spirals about the center of the cavity. This paper applies a quartic twist theory recently proposed to describe the elasticity of chromonics to explain a series of inversion rings observed in aqueous solutions of SSY at two different concentrations. The theory features a phenomenological length a, whose measure is extracted from the data and shown to be fairly independent of the cavity radius, as expected for a material constant.

In this paper, we model the configurations of a system of hard rods by viewing each rod in a cell formed by its neighbors. By minimizing the free energy in the model and performing molecular dynamics, where, in both cases, the shape of the cell is a free parameter, we obtain the equilibrium orientational order parameter, free energy and pressure of the system. Our model enables the calculation of anisotropic stresses exerted on the walls of the cell due to shape change of the rod in photoisomerization. These results are a key step towards understanding molecular shape change effects in photomechanical systems under illumination., Comment: 15 pages, 8 figures

The elastic theory of chromonic liquid crystals is not completely established. We know, for example, that for anomalously low twist constants (needed for chromonics) the classical Oseen-Frank theory may entail paradoxical consequences when applied to describe the equilibrium shapes of droplets surrounded by an isotropic phase: contrary to experimental evidence, they are predicted to dissolve in a plethora of unstable smaller droplets. We proposed a quartic twist theory that prevents such an instability from happening. Here, we apply this theory to the data of two experiments devised to measure the planar anchoring strength at the plates bounding a twist cell filled with a chromonic liquid crystal; these data had before been interpreted within the Oseen-Frank theory. We show that the quartic twist theory affords a better agreement with the experimental data, while delivering in one case a larger value for the anchoring strength.

This paper, originally motivated by a question raised by Wood and Hanna [Soft Matter, 15, 2411 (2019)], shows that pure measures of bending for soft plates can be defined by introducing the class of bending-neutral deformations, finite incremental changes of the plate's shape bearing no further bending. This class of deformations is subject to a geometric compatibility condition, which is fully characterized. A tensorial pure measure of bending, which is accordingly invariant under bending-neutral deformations, is described in details. As shown by an illustrative class of examples, the general notion of pure measure of bending could be of use to formulate direct theories for soft plates, where stretching and bending energies are kept separate.

Chromonic nematics are lyotropic liquid crystals that have already been known for half a century, but have only recently raised interest for their potential applications in life sciences. Determining elastic constants and anchoring strengths for rigid substrates has thus become a priority in the characterization of these materials. Here, we present a method to determine chromonics' planar anchoring strength. We call it geometric as it is based on recognition and fitting of the stable equilibrium shapes of droplets surrounded by the isotropic phase in a thin cell with plates enforcing parallel alignments of the nematic director. We apply our method to shapes observed in experiments; they resemble elongated rods with round ends, which are called batonnets. Our theory also predicts other droplets' equilibrium shapes, which are either slender and round, called discoids, or slender and pointed, called tactoids. In particular, sufficiently small droplets are expected to display shape bistability, with two equilibrium shapes, one tactoid and one discoid, exchanging roles as stable and metastable shapes upon varying their common area.

In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential, a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.

Frustration in nematic-ordered media (endowed with a director field) is treated in a purely geometric fashion in a flat, two-dimensional space. We recall the definition of quasi-uniform distortions and envision these as viable ways to relieve director fields prescribed on either a straight line or the unit circle. We prove that using a planar spiral is the only way to fill the whole plane with a quasi-uniform distortion. Apart from that, all relieving quasi-uniform distortions can at most be defined in a half-plane; however, in a generic sense, they are all asymptotically spirals.

An elastic quartic twist theory has recently been proposed for chromonic liquid crystals, intended to overcome the paradoxical conclusions encountered by the classical Oseen-Frank theory when applied to droplets submerged in an isotropic fluid environment. However, available experimental data for chromonics confined to cylindrical cavities with degenerate planar anchoring on their lateral boundary can be explained equally well by both competing theories. This paper identifies a means to differentiate these theories both qualitatively and quantitatively. They are shown to predict quite different core defects for the twisted hedgehogs that chromonics generate when confined to a fixed spherical cavity with homeotropic anchoring. In the quartic twist theory, the defect core is estimated to be nearly one order of magnitude larger (tens of microns) than in the other and, correspondingly, the director field lines describe Archimedean spirals instead of logarithmic ones., Comment: arXiv admin note: text overlap with arXiv:2211.15820

Chromonic liquid crystals are lyotropic materials which are attracting growing interest for their adapatbility to living systems. To describe their elastic properties, the classical Oseen-Frank theory requires anomalously small twist constants and (comparatively) large saddle-splay constants, so large as to violate one of Ericksen's inequalities, which guarantee that the Oseen-Frank stored-energy density is bounded below. While such a violation does not prevent the existence and stability of equilibrium distortions in problems with fixed geometric confinement, the study of free-boundary problems for droplets has revealed a number of paradoxical consequences. Minimizing sequences driving the total energy to negative infinity have been constructed by employing ever growing needle-shaped tactoids incorporating a diverging twist [Phy. Rev. E 106, 044703 (2022)]. To overcome these difficulties, we propose here a novel elastic theory that extends for chromonics the classical Oseen-Frank stored energy by adding a quartic twist term. We show that the total energy of droplets is bounded below in the quartic twist theory, so that the known paradoxes are ruled out. The quartic term introduces a phenomenological length $a$ in the theory; this affects the equilibrium of chromonics confined within capillary tubes. Use of published experimental data allows us to estimate $a$.

We study the spontaneous out-of-plane bending of a planar untwisted ribbon composed of nematic polymer networks activated by a change in temperature. Our theory accounts for both stretching and bending energies, which compete to establish equilibrium. We show that when equilibrium is attained these energy components obey a \emph{complementarity} relation: one is maximum where the other is minimum. Moreover, we identify a \emph{bleaching} regime: for sufficiently large values of an activation parameter (which measures the mismatch between the degrees of order in polymer organization in the reference and current configurations), the ribbon's deformation is essentially independent of its thickness., Comment: Final published version

We study the equilibria of a photoresponsive nematic elastomer ribbon within a continuum theory that builds upon the statistical mechanics model put forward by Corbett and Warner [Phys. Rev. E 78, 061701 (2008)]. We prove that the spontaneous deformation induced by illumination is not monotonically dependent on the intensity $I$. The ribbon's deflection first increases with increasing $I$, as expected, but then decreases and abruptly ceases altogether at a critical value $I_\mathrm{e}$ of $I$. $I_\mathrm{e}$, which is enclosed within a hysteresis loop, marks a first-order shape transition. Finally, we find that there is a critical value of the ribbon's length, depending only on the degree of cross-linking in the material, below which no deflection can be induced in the ribbon, no matter how intense is the light shone on it.

Chromonic liquid crystals constitute a novel lyotropic phase, whose elastic properties have so far been modeled within the classical Oseen-Frank theory, provided that the twist constant is assumed to be considerably smaller than the saddle-splay constant, in violation of one Ericksen inequality. This paper shows that paradoxical consequences follow from such a violation for droplets of these materials surrounded by an isotropic fluid. For example, tactoids with a degenerate planar anchoring simply disintegrate indefinitely in myriads of smaller ones.

This paper proposes a theory that bridges classical analytical mechanics and nonequilibrium thermodynamics. Its intent is to derive the evolution equations of a system from a stationarity principle for a suitably augmented Lagrangian action. This aim is attained for homogeneous systems, described by a finite number of state variables depending on time only. In particular, it is shown that away from equilibrium free energy and entropy are independent constitutive functions.

The ground state of chromonic liquid crystals, as revealed by a number of recent experiments, is quite different from that of ordinary nematic liquid crystals: it is twisted instead of uniform. The common explanation provided for this state within the classical elastic theory of Frank demands that one Ericksen's inequality is violated. Since in general such a violation makes Frank's elastic free-energy functional unbounded below, the question arises as to whether the twisted ground state can be locally stable. We answer this question in the affirmative. In reaching this conclusion, a central role is played by the specific boundary conditions imposed in the experiments on the boundary of rigid containers and by a general formula that we derive here for the second variation of Frank's elastic free energy.

We present a theory of deformation of ribbons made of nematic polymer networks (NPNs). These materials exhibit properties of rubber and nematic liquid crystals, and can be activated by external stimuli of heat and light. A two-dimensional energy for a sheet of such a material has already been derived from the celebrated neo-classical energy of nematic elastomers in three space dimensions. Here, we use a dimension reduction method to obtain the appropriate energy for a ribbon from the aforementioned sheet energy. We also present an illustrative example of a rectangular NPN ribbon that undergoes in-plane serpentine deformations upon activation under an appropriate set of boundary conditions., Comment: Published version

Within this work we derive and analyse an expression for the free energy of a single-species system in the thermodynamic limit in terms of a generalised cavity volume, that is exact in general, and in principle applicable to systems across their entire range of density, as well as to particles within a general coordinate space. This provides a universal equation of state, and can thus relate the cavity volume to classical results, such as Mayer's cluster expansions. Through this we are able to provide some insight into the connections between cavity volume and free energy density, as well as their consequences. We use examples which permit explicit computations to further probe these results, reclaiming the exact results for a classical Tonks gas and providing a novel derivation of Onsager's free energy for a single species, isotropic system. Given the complexity of the problem we also provide a local lattice ansatz, exact in one dimension, with which we may approximate the cavity volume for hard sphere systems to provide an accurate equation of state in the cases of hard disks and spheres in both dilute regimes as well as beyond the freezing transition.

We study droplets of chromonic liquid crystals squeezed between parallel plates inducing degenerate tangential anchoring on the nematic director. In the coexistence regime, where droplets in the nematic phase are at equilibrium with the surrounding melt, our two-dimensional theoretical model predicts a regime of shape bistability for sufficiently large bipolar droplets, where tactoids (pointed shapes) and discoids (smooth shapes) coexist in equilibrium. This phenomenon has not yet been observed. Two-dimensional droplets of disodium cromoglycate (DSCG) have been the object of a thorough experimental study [Y.-K. Kim et al., J. Phys.: Condens. Matter 25, 404202 (2013)]. We show that our theory is in good quantitative agreement with these data and extract from them what promises to be a more accurate estimate for the isotropic surface tension at the nematic/melt interface of DSCG.

Nematic polymer networks (NPNs) are nematic elastomers within which the nematic director is enslaved to the elastic deformation. The elastic free energy of a NPN sheet of thickness $h$ has both stretching and bending components (the former scaling like h, the latter scaling like $h^3$). NPN sheets bear a director field $\mathbf{m}$ imprinted in them (usually, uniformly throughout their thickness); they can be activated by changing the nematic order (e.g. by illumination or heating). This paper illustrates an attempt to compute the bending energy of a NPN sheet and to show which role it can play in determining the activated shape. Our approach is approximate: the activated surface consists of flat sectors connected by ridges, where the unit normal jumps and the bending energy is concentrated. By increasing the number of ridges, we should get closer to the real situation, where the activated surface is smooth and the bending energy is distributed on it. The method is applied to a disk with imprinted a spiraling hedgehog. It is shown that upon activation the disk, like a tiny hand, is able to grab a rigid lamina.

We study packings of hard spheres on lattices. The partition function, and therefore the pressure, may be written solely in terms of the accessible free volume, i.e. the volume of space that a sphere can explore without touching another sphere. We compute these free volumes using a leaky cell model, in which the accessible space accounts for the possibility that spheres may escape from the local cage of lattice neighbors. We describe how elementary geometry may be used to calculate the free volume exactly for this leaky cell model in two- and three-dimensional lattice packings and compare the results to the well-known Carnahan-Starling and Percus-Yevick liquid models. We provide formulas for the free volumes of various lattices and use the common tangent construction to identify several phase transitions between them in the leaky cell regime, indicating the possibility of coexistence in crystalline materials., Comment: 18 pages, 10 figures

Tactoids are pointed, spindle-like droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a novel class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primarily among them). We show how the prevalence of a shape is affected by the drop's volume $V_0$ and the saddle-splay constant $K_{24}$ of the material. Tactoids, in particular, prevail for small $V_0$ and small $K_{24}$ (appropriately scaled). Our class of shapes (and director fields) is sufficiently different from those employed so far to unveil a rather different role of $K_{24}$.

Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put our theory to the test by studying the spontaneous deformation of a disk on which a radial hedgehog was imprinted at the time of crosslinking. We predict the number of folds that develop in terms of the degree of order induced in the material by external agents (such as heat and illumination).

The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its attribution has been questioned, and recent rigorous dimension-reduction tools (based on $\Gamma$-convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate's mid surface (alongside the expected measures of bending). The incompatibility with $\Gamma$-convergence also appears to be removed in the cases where contact with that method and ours can be made.

The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director $\mathbf{n}$. The ground state is attained when $\mathbf{n}$ is uniform in space; all other states, which have a non-vanishing gradient $\nabla\mathbf{n}$, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from $\mathbf{n}$ and $\nabla\mathbf{n}$ a third-rank, symmetric and traceless tensor $\mathbf{A}$ (the octupolar tensor). The (nonlinear) eigenvectors of $\mathbf{A}$ associated with the local maxima of its cubic form $\Phi$ on the unit sphere (its octupolar potential) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape, but differently inflated.

Ordinary nematic liquid crystals are characterized by having a uniform director field as ground state. In such a state, the director is the same everywhere and no distortion is to be seen at all. We give a definition of uniform distortion which makes precise the intuitive notion of seeing everywhere the same director landscape. We characterize all such distortions and prove that they fall into two families, each described by two scalar parameters. Uniform distortions exhaust R. Meyer's heliconical structures, which, as it has recently been recognized, include the ground state of twist-bend nematics. The generalized elasticity of these new phases is treated with a simple free-energy density, which can be minimized by both uniform and non-uniform distortions, the latter injecting a germ of elastic frustration.

Octupolar tensors are third order, completely symmetric and traceless tensors. Whereas in 2D an octupolar tensor has the same symmetries as an equilateral triangle and can ultimately be identified with a vector in the plane, the symmetries that it enjoys in 3D are quite different, and only exceptionally reduce to those of a regular tetrahedron. By use of the octupolar potential that is, the cubic form associated on the unit sphere with an octupolar tensor, we shall classify all inequivalent octupolar symmetries. This is a mathematical study which also reviews and incorporates some previous, less systematic attempts.

Toroidal nematics are droplets of nematic liquid crystals in the form of a circular torus. When the nematic director is subject to planar degenerate boundary conditions, the bend-only director field with vector lines along the parallels of all internal torodial shells is an equilibrium solution for all values of the elastic constants. Local stability analyses have shown that an instability is expected to occur for sufficiently small values of the twist elastic constant. It is natural to conjecture that in this regime the global equilibrium would be characterized by a maximum twist deflection on the boundary of the torus, with a twist distribution over the torus' cross-section represented by a fennel-like surface. We prove that surprisingly the stable twist distribution is instead represented by a lily-like surface. Thus the overall maximum twist deflection falls well within the torus. To cope with the complexity of the elastic free-energy functional in the fully non-linear setting, we developed an ad hoc deep-learning optimization method, which here is also duly validated and documented for it promises to be applicable to other similar problems, equally intractable analytically.

Both repulsive and attractive molecular interactions can be used to explain the onset of nematic order. The object of this paper is to combine these two nematogenic molecular interactions in a unified theory. This attempt is not unprecedented; what is perhaps new is the focus on the understanding of nematics in the high density limit. There, the orientational probability distribution is shown to exhibit a unique feature: it has compact support on configuration space. As attractive interactions are turned on, the behavior changes, and at a critical attractive interaction strength, thermotropic behavior of the Maier-Saupe type is attained., Comment: 14 pages, 4 figures

Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planar degenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic environment or cavities carved out of a solid substrate. A universal solution of Frank's elastic free energy is an equilibrium configuration for the nematic director field, irrespective of the values of the elastic constants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of this configuration and identify a range of parameters where the main drive towards instability does not come from the surface-like elastic constant $K_{24}$ being large, but from the the ratio $K_2/K_3$ of the twist to bend elastic constants being small, which also makes our study relevant to chromonic liquid crystals.

When a material surface is functionalized so as to acquire some type of order, functionalization of which soft condensed matter systems have recently provided many interesting examples, the modeller faces an alternative. Either the order is described on the curved, physical surface where it belongs, or it is described on a flat surface that is unrolled as pre-image of the physical surface under a suitable height function. This paper proposes a general method that pursues the latter avenue by lifting whatever order tensor is deemed appropriate from a flat to a curved surface. To produce a specific application, we specialize this method to nematic shells, for which it also provides a simple, but convincing interpretation of the outcomes of some molecular-dynamics experiments on ellipsoidal shells.

Onsager's paper on phase transition and phase coexistence in anisotropic colloidal systems is a landmark in the theory of lyotropic liquid crystals. However, an uncompromising scrutiny of Onsager's original derivation reveals that it would be rigorously valid only for ludicrous values of the system's number density (of the order of the reciprocal of the number of particles) Based on Penrose's tree identity and an appropriate variant of the mean-field approach for purely repulsive, hard-core interactions, our theory shows that Onsager's theory is indeed valid for a reasonable range of densities.

Nematic shells are colloidal particles coated with nematic liquid crystal molecules which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. We describe the nematic order on a shell by a unit director field on an orientable surface. Equilibrium fields can then be found by minimising the elastic energy, which in general is a function of the surface gradient of the director field. We learn how to extract systematically out of this energy a fossil component, related only to the surface and its curvatures, which expresses a curvature potential for the molecular torque. At hyperbolic points on the colloid's surface, and only there, the alignment preferred by the curvature potential may fail to be a direction of principal curvature. There the fossil energy becomes bistable.

Paradoxes in the impact dynamics of rigid bodies are known to arise in the presence of friction. We show here that, on specificc occasions, in the absence of friction, the conservation laws of classical mechanics are also incompatible with the collisions of smooth, strictly convex rigid bodies. Under the assumption that the impact impulse is along the normal direction to the surface at the contact point, two convex rigid bodies which are well separated can come into contact, and then interpenetrate each other. This paradox can be constructed in both 2D and 3D when the collisions are tangential, in which case no momentum or energy transfer between the two bodies is possible. The postcollisional interpenetration can be realized through the contact points or through neighboring points only. The penetration distance is shown to be O(t^3). The conclusion is that rigid body dynamics is not compatible with the conservation laws of classical mechanics.

The celebrated work of Onsager on hard particle systems, based on the truncated second order virial expansion, is valid at relatively low volume fractions for large aspect ratio particles. While it predicts the isotropic-nematic phase transition, it fails to provide a realistic equation of state in that the pressure remains finite for arbitrarily high densities. In this work, we derive a mean field density functional form of the Helmholtz free energy for nematics with hard core repulsion. In addition to predicting the isotropic-nematic transition, the model provides a more realistic equation of state. The energy landscape is much richer, and the orientational probability distribution function in the nematic phase possesses a unique feature: it vanishes on a nonzero measure set in orientational space.

A third-order three-dimensional symmetric traceless tensor, called the \emph{octupolar} tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar \emph{potential}, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima capturing the most probable molecular orientations (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with \emph{three} maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a \emph{separatrix} surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible \emph{intra-octupolar} transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction. Some other properties of octupolar tensors are also studied.

In this experimental and theoretical study, we examine the equilibrium shapes of quasitwo-dimensional twist-bend nematic (Ntb) drops formed within a planarly aligned nematic layer of the liquid crystal CB7CB. Initially, at the setting point of the Ntb phase, the drops assume a nonequilibrium cusped elliptical geometry with the major axis orthogonal to the director of the surrounding nematic fluid; this growth is governed principally by anisotropic heat diffusion. The drops attain equilibrium through thermally driven dynamical evolutions close to their melting temperature. They are associated with a characteristic twin-striped morphology that transforms into the familiar focal conic texture as the temperature is lowered. At equilibrium, large millimetric drops are tactoidlike, elongated along the director of the surrounding nematic fluid. This geometry is explained by a mathematical model that features two dimensionless parameters, of which one is the structural cone angle of the Ntb phase and the other, the relative strength of mismatch elastic energy at the drop's interface. Both parameters are extracted from the observations by measuring the aspect ratio of the equilibrium shapes and the inner corner angle of the cusps.

We represent explicitly the excluded volume Ve{B1,B2} of two generic cylindrically symmetric, convex rigid bodies, B1 and B2, in terms of a family of shape functionals evaluated separately on B1 and B2. We show that Ve{B1,B2} fails systematically to feature a dipolar component, thus making illusory the assignment of any shape dipole to a tapered body in this class. The method proposed here is applied to cones and validated by a shape-reconstruction algorithm. It is further applied to spheroids (ellipsoids of revolution), for which it shows how some analytic estimates already regarded as classics should indeed be emended.

Octupolar order is described in two space dimensions in terms of the maxima (and conjugated minima) of the probability density associated with a third-rank, fully symmetric and traceless tensor. Such a representation is shown to be equivalent to diagonalizing the relevant third-rank tensor, an equivalence which however is only valid in the two-dimensional case.